He wants to put on kohl on Fridays and his parents don’t want him to do that

It is not Sunnah to put on kohl on Fridays. It was not the practice of the Prophet (blessings and peace of Allah be upon him) to put on kohl on this day in particular. Rather what is narrated in the Sunnah is that the Prophet (blessings and peace of Allah be upon him) put on kohl and encouraged others to put on kohl because it is beneficial for the eyes, without restricting it to any particular day, whether Friday or any other day. 

For a man to put on kohl as an adornment is something that is not acceptable in most societies. Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was uncertain as to whether it is permissible; see that in the answer referred to above. 

Some negative consequences may result from doing it; people may think badly of the one who does that and some people may find it weird. It is not appropriate for a Muslim to put himself in a position where he will be thought badly of. 

If we add to that your parents’ objection to this action, then you should obey them, because they are doing that out of love for their child and the wish to keep anything that may adversely affect him away from him. 

And Allah knows best.